Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems

Abstract

A novel swarm intelligence optimization technique is proposed called dragonfly algorithm (DA). The main inspiration of the DA algorithm originates from the static and dynamic swarming behaviours of dragonflies in nature. Two essential phases of optimization, exploration and exploitation, are designed by modelling the social interaction of dragonflies in navigating, searching for foods, and avoiding enemies when swarming dynamically or statistically. The paper also considers the proposal of binary and multi-objective versions of DA called binary DA (BDA) and multi-objective DA (MODA), respectively. The proposed algorithms are benchmarked by several mathematical test functions and one real case study qualitatively and quantitatively. The results of DA and BDA prove that the proposed algorithms are able to improve the initial random population for a given problem, converge towards the global optimum, and provide very competitive results compared to other well-known algorithms in the literature. The results of MODA also show that this algorithm tends to find very accurate approximations of Pareto optimal solutions with high uniform distribution for multi-objective problems. The set of designs obtained for the submarine propeller design problem demonstrate the merits of MODA in solving challenging real problems with unknown true Pareto optimal front as well. Note that the source codes of the DA, BDA, and MODA algorithms are publicly available at http://www.alimirjalili.com/DA.html.

Appendices

Appendix 1: Single-objective test problems utilized in this work

See Tables 10, 11, 12.

Table 10 Unimodal benchmark functions

Table 11 Multimodal benchmark functions

Table 12 Composite benchmark functions

Appendix 2: Multi-objective test problems utilized in this work

ZDT1:

$${\text{Minimise}}:f_{1} \left( x \right) = x_{1}$$

(7.1)

$${\text{Minimise}}:f_{2} \left( x \right) = g\left( x \right) \times h\left( {f_{1} \left( x \right), g\left( x \right)} \right)$$

(7.2)

$${\text{Where}}:G\left( x \right) = 1 + \frac{9}{N - 1}\mathop \sum \limits_{i = 2}^{N} x_{i}$$

(7.3)

$$h\left( {f_{1} \left( x \right),g\left( x \right)} \right) = 1 - \sqrt {\frac{{f_{1} \left( x \right)}}{g\left( x \right)}} \quad 0 \le x_{i} \le 1, 1 \le i \le 30$$

(7.4)

ZDT2:

$${\text{Minimise}}:f_{1} \left( x \right) = x_{1}$$

(7.5)

$${\text{Minimise}}:f_{2} \left( x \right) = g\left( x \right) \times h\left( {f_{1} \left( x \right), g\left( x \right)} \right)$$

(7.6)

$${\text{Where}}:G\left( x \right) = 1 + \frac{9}{N - 1}\mathop \sum \limits_{i = 2}^{N} x_{i}$$

(7.7)

$$h\left( {f_{1} \left( x \right),g\left( x \right)} \right) = 1 - \left( {\frac{{f_{1} \left( x \right)}}{g\left( x \right)}} \right)^{2} \quad 0 \le x_{i} \le 1, 1 \le i \le 30$$

(7.8)

ZDT3:

$${\text{Minimise}}:f_{1} \left( x \right) = x_{1}$$

(7.9)

$${\text{Minimise}}:f_{2} \left( x \right) = g\left( x \right) \times h\left( {f_{1} \left( x \right), g\left( x \right)} \right)$$

(7.10)

$${\text{Where}}:G\left( x \right) = 1 + \frac{9}{29}\mathop \sum \limits_{i = 2}^{N} x_{i}$$

(7.11)

$$h\left( {f_{1} \left( x \right),g\left( x \right)} \right) = 1 - \sqrt {\frac{{f_{1} \left( x \right)}}{g\left( x \right)}} - \left( {\frac{{f_{1} \left( x \right)}}{g\left( x \right)}} \right)\text{sin}\left( {10\pi f_{1} \left( x \right)} \right)\quad 0 \le x_{i} \le 1, 1 \le i \le 30$$

(7.12)

ZDT1 with linear PF:

$${\text{Minimise}}:f_{1} \left( x \right) = x_{1}$$

(7.13)

$${\text{Minimise}}:f_{2} \left( x \right) = g\left( x \right) \times h\left( {f_{1} \left( x \right), g\left( x \right)} \right)$$

(7.14)

$${\text{Where}}:G\left( x \right) = 1 + \frac{9}{N - 1}\mathop \sum \limits_{i = 2}^{N} x_{i}$$

(7.15)

$$h\left( {f_{1} \left( x \right),g\left( x \right)} \right) = 1 - \frac{{f_{1} \left( x \right)}}{g\left( x \right)}\quad 0 \le x_{i} \le 1, 1 \le i \le 30$$

(7.16)

ZDT2 with three objectives:

$${\text{Minimise}}:f_{1} \left( x \right) = x_{1}$$

(7.17)

$${\text{Minimise}}:f_{2} \left( x \right) = x_{2}$$

(7.18)

$${\text{Minimise}}:f_{3} \left( x \right) = g\left( x \right) \times h\left( {f_{1} \left( x \right), g\left( x \right)} \right) \times h\left( {f_{2} \left( x \right), g\left( x \right)} \right)$$

(7.19)

$${\text{Where}}:G\left( x \right) = 1 + \frac{9}{N - 1}\mathop \sum \limits_{i = 3}^{N} x_{i}$$

(7.20)

$$h\left( {f_{1} \left( x \right),g\left( x \right)} \right) = 1 - \left( {\frac{{f_{1} \left( x \right)}}{g\left( x \right)}} \right)^{2} \quad 0 \le x_{i} \le 1, 1 \le i \le 30$$

(7.21)